On the properties of canonical solution operator to ${\bar{\partial }}$ restricted to Dirichlet space

Abstract

In this paper, we first show that the canonical solution operator $S_1$ to ${\bar{\partial }}$ restricted to (0,1)-forms with holomorphic function coefficients can be expressed by an integral operator using the Dirichlet kernel. Then we prove that operator $S_k\,(k\ge 1)$ is a Hilbert–Schmidt operator on the Dirichlet space of ${\mathbb {D}}$, but fails to be the Hilbert–Schmidt operator on the Dirichlet space of ${\mathbb {D}}^{2}$. Finally we show that the concomitant operator $P_{k}$ of $S_{k}\,(k\ge 2)$ is similar to the direct sum of k copies of the concomitant operator $P_{1}$ of $S_{1}$.